Another temperament question -- and some remade music

Hi all.

A few days ago, I was looking for specific properties in 7-limit temperaments and, so far, superpyth and meantone were the two which had a very special thing in common, as far as my wish to approximate 4:5:6:7 was concerned. It was the fact that when I make the 6/5 mistuned by just the same amount as 5/4, then 7/6 is not more out of tune than 3/2 -- this is true for both C-E-G-A# in meantone and C-D#-G-Bb in superpyth. Do you think there could be even more temperaments meeting this condition? I would be very interested to learn about them.

Thanks in advance.

Petr

PS: Some of you may remember my superpyth piece which I sent recently. Now I've made it longer so if you want to hear the newer version, here it is: http://download.yousendit.com/9CA25E666D053FA9

--- In tuning@yahoogroups.com, Petr Pařízek <p.parizek@...> wrote:
>
> Hi all.
>
> A few days ago, I was looking for specific properties in 7-limit
> temperaments and, so far, superpyth and meantone were the two
> which had a very special thing in common, as far as my wish to
> approximate 4:5:6:7 was concerned. It was the fact that when I
> make the 6/5 mistuned by just the same amount as 5/4, then 7/6
> is not more out of tune than 3/2 -- this is true for both
> C-E-G-A# in meantone and C-D#-G-Bb in superpyth. Do you think
> there could be even more temperaments meeting this condition?
> I would be very interested to learn about them.

Hi Petr- May I ask why this is important?

> PS: Some of you may remember my superpyth piece which I sent
> recently. Now I've made it longer so if you want to hear the
> newer version, here it is:
> http://download.yousendit.com/9CA25E666D053FA9

Nice!

-Carl

Carl wrote:

> Hi Petr- May I ask why this is important?

I'm not sure what you mean. I was just looking for good 7-limit temperaments and this question of "mistuning in-between" which I've mentioned seemed to play an important part.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Carl wrote:
>
> > Hi Petr- May I ask why this is important?
>
> I'm not sure what you mean. I was just looking for good 7-limit
> temperaments and this question of "mistuning in-between" which
> I've mentioned seemed to play an important part.
>
> Petr

What makes a 7-limit temperament good? Isn't it just low error
and being able to make many consonances with few notes?

I mean, I don't know why you're singling out 6/5, 5/4, 3/2, and
7/6 and not looking at 7/4 or 7/5. And I don't know why having
equal mistuning in the way you describe is desirable.

-Carl

31 is a good meantone temperment of the 1-3-5-7, if you want such a thing.
but didn't Erlich based much of his work on 22 on how this tetrad worked in it?

,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
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-----Original Message-----
From: Carl Lumma [mailto:carl@lumma.org]
Sent: Monday, February 4, 2008 02:02 PM
To: tuning@yahoogroups.com
Subject: [tuning] Re: Another temperament question -- and some remade music

--- In tuning@yahoogroups.com, Petr Par�zek <p.parizek@...> wrote:
>
> Carl wrote:
>
> > Hi Petr- May I ask why this is important?
>
> I'm not sure what you mean. I was just looking for good 7-limit
> temperaments and this question of "mistuning in-between" which
> I've mentioned seemed to play an important part.
>
> Petr

What makes a 7-limit temperament good? Isn't it just low error
and being able to make many consonances with few notes?

I mean, I don't know why you're singling out 6/5, 5/4, 3/2, and
7/6 and not looking at 7/4 or 7/5. And I don't know why having
equal mistuning in the way you describe is desirable.

-Carl

--- In tuning@yahoogroups.com, kraiggrady@... wrote:
>
> 31 is a good meantone temperment of the 1-3-5-7, if you want
> such a thing.
> but didn't Erlich based much of his work on 22 on how this
> tetrad worked in it?

The centerpiece of Paul's early work are his scales in 22,
which are a lot like the diatonic scale in meantone but with
7-limit tetrads instead of triads.

Later, Paul contributed heavily to our understanding of
regular temperaments, of which his scales in 22 are only one
example.

-Carl

Carl wrlte:

> I mean, I don't know why you're singling out 6/5, 5/4, 3/2, and
> 7/6 and not looking at 7/4 or 7/5.

As far as my own experience goes, if I play some sort of tempered 4:5:6:7 with a sharp timbre, sometimes the beats in the epimoric factors get slightly more pronounced than the beats in the other ones. That's why I prefer slower beats in 7/6 instead of having slow beats in 7/5 and fast beats in 7/6.

> And I don't know why having equal mistuning in the way you describe is desirable.

Because some intervals can get even closer to JI then. In 1/4-comma or 1/3-comma meantone, for example, one of the thirds is pure while the other one is tempered by 1/4 or 1/3-comma, respectively. Since both 6/5 and 5/4 are higher in the harmonic series than 4/3 or 3/2, their guide tones are also higher and therefore their beats are faster. But in 2/7-comma meantone, both thirds are tempered by only 1/7-comma so their beats don't get so fast. The same goes for the superpyth which I used on the recording where the generator is (3200/3)^(1/17). This makes both C-D# and D#-G larger than 5/4 and 6/5 by only 1/17 of the 20480/19683, which is significantly less than in the case of using the 9th root of 40 where D#-G is larger than 6/5 by 1/9 of that interval.

Petr

Petr wrote...

> > I mean, I don't know why you're singling out 6/5, 5/4, 3/2, and
> > 7/6 and not looking at 7/4 or 7/5.
>
> As far as my own experience goes, if I play some sort of tempered
> 4:5:6:7 with a sharp timbre, sometimes the beats in the epimoric
> factors get slightly more pronounced than the beats in the other
> ones. That's why I prefer slower beats in 7/6 instead of having
> slow beats in 7/5 and fast beats in 7/6.

OK.

> > And I don't know why having equal mistuning in the way you
> > describe is desirable.
>
> Because some intervals can get even closer to JI then. In
> 1/4-comma or 1/3-comma meantone, for example, one of the thirds
> is pure while the other one is tempered by 1/4 or 1/3-comma,
> respectively. Since both 6/5 and 5/4 are higher in the harmonic
> series than 4/3 or 3/2, their guide tones are also higher and
> therefore their beats are faster. But in 2/7-comma meantone,
> both thirds are tempered by only 1/7-comma so their beats don't
> get so fast. The same goes for the superpyth which I used on the
> recording where the generator is (3200/3)^(1/17).

This generator is very close to a generator of 7-limit TOP
superpyth. Are you using it with a pure octave?

By the way, I found your notation here confusing. You mean
3200 cents (right?), so you don't want the 17th root. Rather,
just 3200/51.

As far as allowing less mistuning on epimoric ratios, I'll
have to think about that. The usual approach has been to
preferentially reduce the mistuning of *simple* ratios, which
favors 3/2, 5/4, and 7/4 over 6/5, 7/5, and 7/6.

It sounds like your overall goal is to minimize the beat rates
of your intervals. Is that right? If so, I'm not aware of
any direct attempt to solve that for 7-limit linear
temperaments. It is an interesting proposition.

-Carl

Hi Carl,

though I'm back on the list now, I won't repost my previous message as I've successfully deleted it so I'll respond to your newest words now.

> I don't see how epimores / guide tones are important here.
> Isn't it just that smaller intervals are liable to beat faster
> when mistuned by a fixed amount?

I don't know how to explain why it sounds to me like the beats in the epimoric intervals were somewhat clearer sometimes but I've experienced this more times. I'll have to play around with it to see if I can reach any meaningful conclusion. As to the guide tones, their beats get faster as their pitches get higher. If you play 4:5:6, then the guide tone of the major third is 20 and the one for the minor third is 30 (which is 50% higher). If you leave the fifth pure and shift the middle tone either slightly up or down, then the minor third will beat 50% faster than the major third.

> In that case, I'd suggest
> what might be called "inverse Tenney weighting". In any
> harmonic limit, the most complex ratios are the smallest.
> So whereas Tenney weighting preferentially distributes error
> to the simple intervals, you want to distribute it to the
> complex ones.

Sounds interesting.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Hi Carl,
>
> though I'm back on the list now, I won't repost my previous
> message as I've successfully deleted it so I'll respond to your
> newest words now.
>
> > I don't see how epimores / guide tones are important here.
> > Isn't it just that smaller intervals are liable to beat faster
> > when mistuned by a fixed amount?
>
> I don't know how to explain why it sounds to me like the beats in
> the epimoric intervals were somewhat clearer sometimes but I've
> experienced this more times.

For what it's worth, epimores are by nature the smallest intervals
of a given limit.

> I'll have to play around with it to
> see if I can reach any meaningful conclusion.

Let us know what you find.

-Carl